[I know it's is a Physics Problem but I am sure that I am committing mistake in integration part. That's why so posting here. I am new to calculus-based physics so I frequently make conceptual mistakes when setting up integrals; I would really appreciate someone pointing them out.]
Goal: Finding the center of mass of a semicircular wire/disk of on non-negligible width, with the inner radius being $R_1$ and outer radius being $R_2$.
I am gonna start this with a goal of setting up a Reimann Sum. First I divide the "arc" of angle $\pi$ into $n$ sub-arcs of equal angle $\Delta\theta$
The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a height, $H_i$, approximating the height of the center of mass of each sub arc, hoping that the error goes to $0$ in the limit as $n \to \infty$, and multiply this by the mass of the sub arc. Pushing this through the limiting process, I set up the integral of $H$ w.r.t $m$.
Finding $H_i$: Now, as $\Delta\theta \to0$, the sector-difference region formed by each sub-arc should get closer and closer to a tilted rectangle. Assuming that to be $\color{green}{\text{True}}$, the center of mass of each sub-arc (being approximated by a titled rectangle) would be a distance $H_i=\frac{(R_1+R_2)\sin(θ)}{2}$ above the origin.
Lastly, since the shape has a constant mass per unit area, the differential mass and total mass can be replaced by differential area and total area. Using the sector area formula for each subinterval, the differential area, $dA$, should be equal to $0.5\;d\theta (R_2^2-R_1^2)$
Solving this gives me $$y_{com}=\frac{(R1+R2)}{\pi}$$ which upon looking up is clearly wrong. It is interesting thought that it gives the correct result when $R_1=R_2$ ($0$-thickness). What is the error in my reasoning?